Some Notes On The Product Space Theory And Its Application In Mind

Since 1969, K. Nagami studied the product of two paracompact space, it has been significant improvement that the research of the topological spaces'product which was described by covering . Especially in recent years, K.Chiba,Y. Yajima and pres peiyong-zhu have obtained many very many and good results about Tychonoff product,σ?product and the inverse limit of two or infinite paracompact space ,metacompact space,submetacompact spaces which were described by covering. So it is a natural question that the following issues are raised.Question: does the topological space class ,such as strongly paracompact and ultr -aparacompact which are stronger than paracompact and are described by covering have the results of product similar to the weak coverage?This article is mainly to study this issue.Two equivalent characterizations about the Tychonoff product of ultraparacompact and genetic ultraparacompact space class are as follows:1) let X =Πσ∈ΣXσbeλ-ultraparacompact (hereditarily ultraparacompact) space, then it is ultraparacompact (hereditarily ultraparacompact) space if and only ifΠσ∈F Xσis (hereditarily ultraparacompact) ultraparacompact space for every F∈[Σ]<ω.2) Forω- ultraparacompact (hereditarily ultraparacompact) space. X =Πσ∈ωXσ,the followings are equivalent:(1) X is ultraparacompact (hereditarily ultraparacompact) space;(2) ?F∈[ω]<ω,Πi∈F Xi is ultraparacompact (hereditarily ultraparacompact) space;(3) ?n∈ω,Πi≤n Xi is ultraparacompact (hereditarily ultraparacompact) space;In addition, in this paper some research about product space theory in the appli- cation of topological dynamical systems was done.Two results about generalized periodic points (including the cyclic points, the reunification point, non-wandering points ,ω- limit point, Chain Recurrent Points ect.) of continuous self map were as follows: 3) Let X ,Y be two topological spaces, if the point x∈X is the recurrent point of f : X→X,the point y∈Y is contractible point of g : Y→Y, then the point ( x, y)∈X×Yis the recurrent point of h = f×g : X×Y→X×Y.4) Let {( Xα, fα)}α∈Γis a family of topological dynamical system, if wα∈Xαis neighborhood contractible point for ?α∈Γ, then w = ( wα)α∈Γis nonwandering point of X =Πα∈ΓXα.This results promote the product space and its application to further develop the theory.